truth_conditions - 6 Truth and Possible Worlds We have...

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1 6. Truth and Possible Worlds We have defined logical entailment, consistency, and the connectives , , ¬ , all in terms of belief. In view of the close connection between belief and truth, described in the first section, we should see what each of these has to do with truth. Suppose A is true, and A entails B . Does B then have to be true as well, or could it be false? If it is possible for B to be false here, then that’s bad news. For then even a perfect thinker, starting with truth, could end up in error. Fortunately we shall see that this is not possible. Logical entailment cannot lead from truth to something false. Entailment, we say, is truth preserving . 6.1 Possible Worlds There are many other links between the logical terms and truth. In order to investigate all of these it will be useful to introduce the notion of a possible world . This will allow to define the structure of propositions more precisely than before, and see just what it means for a proposition to be true. Recall the idea of an expansion of an epistemic state – knowledge is added to the state, without removing any. This notion of expansion naturally leads to the idea of a learning path of epistemic states. A learning path is a sequence of epistemic states where each state is an expansion of the previous one. A learning path might be seen as the intellectual biography of a perfect thinker, as she acquires more and more knowledge. One question about learning paths is whether they can ever come to an end. Is there any epistemic state that’s so big it can’t be expanded further? Theoretically, at least, a perfect thinker might get to the stage where she has a firm, definite opinion on every issue. She has unshakable beliefs about the doings of every single beetle in the Amazon basin, knows everything there is to know about every person on the planet, even the number of hairs on their head. Her epistemic state is every bit as detailed and complicated as the real world itself! No human could have such an epistemic state, of course; it’s just a theoretical entity. Such an epistemic state can be called maximal , since it cannot be expanded. It is not contained in any epistemic state (apart from itself of course). We will call such a maximal epistemic state a possible world . Definition A possible world is a maximal epistemic state, i.e. one which cannot be expanded. It is not contained in any other epistemic state. There are many possible worlds, in this sense, as there are many possible combinations of beliefs. A possible world doesn’t have to match the real world, it only has to be a possible state of belief for a perfect thinker. It has to be coherent, make sense, and so on, but it doesn’t have to be true.
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2 The opposite of a maximal epistemic state (possible world) is a minimal epistemic state. This is a state with no information at all, so that it is contained in every epistemic state, even itself. It is clear that there can only be one minimal state. (Why?) Assuming it exists, we will call this state
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truth_conditions - 6 Truth and Possible Worlds We have...

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